Shape maps for second order partial differential equations

TL;DR

This paper introduces shape maps as a geometric tool to analyze singularity formation in solutions of second order PDEs, providing a new perspective on solution behavior and collapse.

Contribution

It develops a geometric framework using jet spaces and linear connections to study shape maps and their evolution, offering new insights into PDE singularities.

Findings

Shape maps relate to solution volume collapse.

Derived evolution equations similar to Raychaudhuri's equation.

Explicit computations and a nontrivial example provided.

Abstract

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example.